As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes.

The “reasons to doubt” that interpretation is that the M2-brane certainly does not support a perturbation theory the way that the superstring does. This is part of the reason why the actual nature of “M-theory” remains mysterious.

Keeping in mind that already string theory itself and in fact already quantum field theory itself have only partially been formulated in a precise way, the conjecture is motivated from the fact that with the available knowledge of these subjects – particularly from duality in string theory – one can see indications that there is a kind of commuting diagram of the form

in some sense. The unknown top left corner here has optimistically been given a name, and that is “M-theory”. But even the rough global structure of the top left corner has remained elusive.

Hints

The available evidence that there is something of interest consists of various facets of the bottom left and the top right entry of the above diagram, that seem to have a common origin in the top left corner.

Membranes

Notably, from the black brane-solution structure in 11-dimensional supergravity and from the brane scan one finds that it contains a 2-brane, called the M2-brane, and to the extent that one has this under control one can show that under “double dimensional reduction” this becomes the string. However, it is clear that this cannot quite give a definition of the top left corner by perturbation theory as the superstringsigma-model does for the bottom left corner, because by the very nature of the conjecture, the top left corner is supposed to be given by a non-perturbative strong-coupling limit of the bottom left corner.

U-duality

Another hint comes from the fact that the U-duality-structure of supergravity theories forms a clear pattern in those dimensions where one understands it well, giving rise to a description of higher dimensional supergravity theories by exceptional generalized geometry. Now, this pattern, as a mathematical pattern, can be continued to the case that would correspond to the top left corner above, by passing to exceptional generalized geometry over hyperbolicKac-Moody Lie algebras such as first E10 and then, ultimately E11. The references there show that these are huge algebraic structures inside which people incrementally find all kinds of relations that are naturally identified with various aspects of M-theory. This leads to the conjecture that M-theory somehow isE11E_{11} in some way. But it all remains rather mysterious at the moment.

In slightly more detail, write, topologically, T2=SA1×SB1T^2 = S^1_A\times S^1_B for the compactification torus of M-theory, where contracting the first SA1S^1_A-factor means passing to type IIA. To obtain type IIB in noncompact 10 dimensions from M-theory, also the second SB1S^1_B is to be compactified (since T-duality sends the radius rAr_A of SA1S^1_A to the inverse radius rB=ℓs2/RAr_B = \ell_s^2 / R_A of SB1S^1_B). Therefore type IIB sugra in d=10d = 10 is obtained from 11d sugra compactified on the torusSA1×SB1S^1_A \times S^1_B. More generally, this torus may be taken to be an elliptic curve and this may vary over the 9d base space as an elliptic fibration.

Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity vacuum is the input data that F-theory is concerned with.

19:33: “Ten years ago we had the embarrassment that there were five consistent string theories plus a close cousin, which was 11-dimensional supergravity.” (19:40): “I promise you that by the end of the talk we have just one big theory.”